Theorem prdssca 17352

Description: Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)

Hypotheses

Ref	Expression
prdsbas.p	⊢ 𝑃 = (𝑆Xs𝑅)
prdsbas.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdsbas.r	⊢ (𝜑 → 𝑅 ∈ 𝑊)

Assertion

Ref	Expression
prdssca	⊢ (𝜑 → 𝑆 = (Scalar‘𝑃))

Proof of Theorem prdssca

Dummy variables 𝑎 𝑐 𝑑 𝑒 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsbas.p	. . . 4 ⊢ 𝑃 = (𝑆Xs𝑅)
2		eqid 2730	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
3		eqidd 2731	. . . 4 ⊢ (𝜑 → dom 𝑅 = dom 𝑅)
4		eqidd 2731	. . . 4 ⊢ (𝜑 → X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) = X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)))
5		eqidd 2731	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥)))))
6		eqidd 2731	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥)))))
7		eqidd 2731	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))) = (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥)))))
8		eqidd 2731	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))) = (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥))))))
9		eqidd 2731	. . . 4 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
10		eqidd 2731	. . . 4 ⊢ (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))})
11		eqidd 2731	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )) = (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )))
12		eqidd 2731	. . . 4 ⊢ (𝜑 → (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))) = (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥))))
13		eqidd 2731	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))) = (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥))))))
14		prdsbas.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
15		prdsbas.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	prdsval 17351	. . 3 ⊢ (𝜑 → 𝑃 = (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
17		eqid 2730	. . 3 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
18		scaid 17211	. . 3 ⊢ Scalar = Slot (Scalar‘ndx)
19		snsstp1 4766	. . . . 5 ⊢ {⟨(Scalar‘ndx), 𝑆⟩} ⊆ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}
20		ssun2 4127	. . . . 5 ⊢ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩} ⊆ ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩})
21	19, 20	sstri 3942	. . . 4 ⊢ {⟨(Scalar‘ndx), 𝑆⟩} ⊆ ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩})
22		ssun1 4126	. . . 4 ⊢ ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ⊆ (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}))
23	21, 22	sstri 3942	. . 3 ⊢ {⟨(Scalar‘ndx), 𝑆⟩} ⊆ (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))⟩, ⟨(+g‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(+g‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(.r‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑅‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓‘𝑥)(le‘(𝑅‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥))), 𝑐 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ (𝑑 ∈ ((2nd ‘𝑎)(𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))𝑐), 𝑒 ∈ ((𝑓 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)), 𝑔 ∈ X𝑥 ∈ dom 𝑅(Base‘(𝑅‘𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓‘𝑥)(Hom ‘(𝑅‘𝑥))(𝑔‘𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑅‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩}))
24	16, 17, 18, 14, 23	prdsbaslem 17349	. 2 ⊢ (𝜑 → (Scalar‘𝑃) = 𝑆)
25	24	eqcomd 2736	1 ⊢ (𝜑 → 𝑆 = (Scalar‘𝑃))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∀wral 3045   ∪ cun 3898   ⊆ wss 3900  {csn 4574  {cpr 4576  {ctp 4578  ⟨cop 4580   class class class wbr 5089  {copab 5151   ↦ cmpt 5170   × cxp 5612  dom cdm 5614  ran crn 5615   ∘ ccom 5618  ‘cfv 6477  (class class class)co 7341   ∈ cmpo 7343  1^st c1st 7914  2^nd c2nd 7915  Xcixp 8816  supcsup 9319  0cc0 10998  ℝ^*cxr 11137   < clt 11138  ndxcnx 17096  Basecbs 17112  +_gcplusg 17153  ._rcmulr 17154  Scalarcsca 17156   ·_𝑠 cvsca 17157  ·_𝑖cip 17158  TopSetcts 17159  lecple 17160  distcds 17162  Hom chom 17164  compcco 17165  TopOpenctopn 17317  ∏_tcpt 17334   Σ_g cgsu 17336  X_scprds 17341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-cnex 11054  ax-resscn 11055  ax-1cn 11056  ax-icn 11057  ax-addcl 11058  ax-addrcl 11059  ax-mulcl 11060  ax-mulrcl 11061  ax-mulcom 11062  ax-addass 11063  ax-mulass 11064  ax-distr 11065  ax-i2m1 11066  ax-1ne0 11067  ax-1rid 11068  ax-rnegex 11069  ax-rrecex 11070  ax-cnre 11071  ax-pre-lttri 11072  ax-pre-lttrn 11073  ax-pre-ltadd 11074  ax-pre-mulgt0 11075

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-map 8747  df-ixp 8817  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-sup 9321  df-pnf 11140  df-mnf 11141  df-xr 11142  df-ltxr 11143  df-le 11144  df-sub 11338  df-neg 11339  df-nn 12118  df-2 12180  df-3 12181  df-4 12182  df-5 12183  df-6 12184  df-7 12185  df-8 12186  df-9 12187  df-n0 12374  df-z 12461  df-dec 12581  df-uz 12725  df-fz 13400  df-struct 17050  df-slot 17085  df-ndx 17097  df-base 17113  df-plusg 17166  df-mulr 17167  df-sca 17169  df-vsca 17170  df-ip 17171  df-tset 17172  df-ple 17173  df-ds 17175  df-hom 17177  df-cco 17178  df-prds 17343

This theorem is referenced by:  pwssca  17392  xpssca  17472  xpsvsca  17473  prdslmodd  20895  dsmmlss  21674  rrxsca  25316